The power transmittance and directivity of light beams propagated along straight circular symmetrical dielectric tubes and emitted from the end surface of the bore are derived from geometrical optics. The radial power distribution in the beam is included as a distribution function <i>F</i>(ρ,θ). The calculations apply to the case of Fresnel reflection losses. Scattering and absorption phenomena and the contribution of rays refracted into the tube wall are not taken into consideration. Numerical results are compared with experimental values obtained with pulsed light beams characterized by an unsymmetric radial power distribiution.
K. VOGEL, "Radiation Characteristics of Light Beams Transmitted Through Straight Dielectric Tubes," J. Opt. Soc. Am. 56, 1222-1226 (1966)